## Tag `02PI`

Chapter 106: Obsolete > Section 106.19: Duplicate and split out references

Lemma 106.19.6. Let $R$ be a local ring.

- If $(M, N, \varphi, \psi)$ is a $2$-periodic complex such that $M$, $N$ have finite length. Then $e_R(M, N, \varphi, \psi) = \text{length}_R(M) - \text{length}_R(N)$.
- If $(M, \varphi, \psi)$ is a $(2, 1)$-periodic complex such that $M$ has finite length. Then $e_R(M, \varphi, \psi) = 0$.
- Suppose that we have a short exact sequence of $2$-periodic complexes $$ 0 \to (M_1, N_1, \varphi_1, \psi_1) \to (M_2, N_2, \varphi_2, \psi_2) \to (M_3, N_3, \varphi_3, \psi_3) \to 0 $$ If two out of three have cohomology modules of finite length so does the third and we have $$ e_R(M_2, N_2, \varphi_2, \psi_2) = e_R(M_1, N_1, \varphi_1, \psi_1) + e_R(M_3, N_3, \varphi_3, \psi_3). $$

Proof.This follows from Chow Homology, Lemmas 41.2.3 and 41.2.4. $\square$

The code snippet corresponding to this tag is a part of the file `obsolete.tex` and is located in lines 2310–2336 (see updates for more information).

```
\begin{lemma}
\label{lemma-periodic-length}
Let $R$ be a local ring.
\begin{enumerate}
\item If $(M, N, \varphi, \psi)$ is a $2$-periodic complex
such that $M$, $N$ have finite length. Then
$e_R(M, N, \varphi, \psi) = \text{length}_R(M) - \text{length}_R(N)$.
\item If $(M, \varphi, \psi)$ is a $(2, 1)$-periodic complex
such that $M$ has finite length. Then
$e_R(M, \varphi, \psi) = 0$.
\item Suppose that we have a short exact sequence of
$2$-periodic complexes
$$
0 \to (M_1, N_1, \varphi_1, \psi_1)
\to (M_2, N_2, \varphi_2, \psi_2)
\to (M_3, N_3, \varphi_3, \psi_3)
\to 0
$$
If two out of three have cohomology modules of finite length so does
the third and we have
$$
e_R(M_2, N_2, \varphi_2, \psi_2) =
e_R(M_1, N_1, \varphi_1, \psi_1) +
e_R(M_3, N_3, \varphi_3, \psi_3).
$$
\end{enumerate}
\end{lemma}
\begin{proof}
This follows from Chow Homology, Lemmas
\ref{chow-lemma-additivity-periodic-length} and
\ref{chow-lemma-finite-periodic-length}.
\end{proof}
```

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